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  2. Generator matrix - Wikipedia

    en.wikipedia.org/wiki/Generator_matrix

    A generator matrix for a linear [,,]-code has format , where n is the length of a codeword, k is the number of information bits (the dimension of C as a vector subspace), d is the minimum distance of the code, and q is size of the finite field, that is, the number of symbols in the alphabet (thus, q = 2 indicates a binary code, etc.).

  3. Group code - Wikipedia

    en.wikipedia.org/wiki/Group_code

    Group code. In coding theory, group codes are a type of code. Group codes consist of linear block codes which are subgroups of , where is a finite Abelian group . A systematic group code is a code over of order defined by homomorphisms which determine the parity check bits. The remaining bits are the information bits themselves.

  4. Reed–Muller code - Wikipedia

    en.wikipedia.org/wiki/Reed–Muller_code

    Traditional Reed–Muller codes are binary codes, which means that messages and codewords are binary strings. When r and m are integers with 0 ≤ r ≤ m, the Reed–Muller code with parameters r and m is denoted as RM ( r , m ). When asked to encode a message consisting of k bits, where holds, the RM ( r , m) code produces a codeword ...

  5. Linear code - Wikipedia

    en.wikipedia.org/wiki/Linear_code

    Linear code. In coding theory, a linear code is an error-correcting code for which any linear combination of codewords is also a codeword. Linear codes are traditionally partitioned into block codes and convolutional codes, although turbo codes can be seen as a hybrid of these two types. [1] Linear codes allow for more efficient encoding and ...

  6. Parity-check matrix - Wikipedia

    en.wikipedia.org/wiki/Parity-check_matrix

    Formally, a parity check matrix H of a linear code C is a generator matrix of the dual code, C ⊥. This means that a codeword c is in C if and only if the matrix-vector product Hc ⊤ = 0 (some authors [1] would write this in an equivalent form, cH ⊤ = 0.) The rows of a parity check matrix are the coefficients of the parity check equations. [2]

  7. Hadamard code - Wikipedia

    en.wikipedia.org/wiki/Hadamard_code

    The Hadamard code is a linear code, and all linear codes can be generated by a generator matrix .This is a matrix such that () = holds for all {,}, where the message is viewed as a row vector and the vector-matrix product is understood in the vector space over the finite field.

  8. Hamming(7,4) - Wikipedia

    en.wikipedia.org/wiki/Hamming(7,4)

    Hamming (7,4) In coding theory, Hamming (7,4) is a linear error-correcting code that encodes four bits of data into seven bits by adding three parity bits. It is a member of a larger family of Hamming codes, but the term Hamming code often refers to this specific code that Richard W. Hamming introduced in 1950.

  9. Singleton bound - Wikipedia

    en.wikipedia.org/wiki/Singleton_bound

    In the linear code case a different proof of the Singleton bound can be obtained by observing that rank of the parity check matrix is . [4] Another simple proof follows from observing that the rows of any generator matrix in standard form have weight at most n − k + 1 {\displaystyle n-k+1} .